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Let \(R\) be a commutative ring and \(M\) be a Noetherian \(R\)-module. The intersection graph of annihilator submodules of \(M\), denoted by \(GA(M)\) is an undirected simple graph whose vertices are the classes of elements of \(Z_R(M)\setminus \text{Ann}_R(M)\), for \(a,b \in R\) two distinct classes \([a]\) and \([b]\) are adjacent if and only if \(\text{Ann}_M(a)\cap \text{Ann}_M(b)\neq 0\). In this paper, we study diameter and girth of \(GA(M)\) and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that \(GA(M)\) is complete if and only if \(Z_R(M)\) is an ideal of \(R\). Also, we show that if \(M\) is a finitely generated \(R\)-module with \(r(\text{Ann}_R(M))\neq \text{Ann}_R(M)\) and \(|m-\text{Ass}_R(M)|=1\) and \(GA(M)\) is a star graph, then \(r(\text{Ann}_R(M))\) is not a prime ideal of \(R\) and \(|V(GA(M))|=|\text{Min}\,\text{Ass}_R(M)|+1\).

A set \(S\) of cycles is minimal unavoidable in a graph family \(\cal{G}\) if each graph \(G \in \cal{G}\) contains a cycle from \(S\) and, for each proper subset \(S^{\prime}\subset S\), there exists an infinite subfamily \(\cal{G}^{\prime}\subseteq\cal{G}\) such that no graph from \(\cal{G}^{\prime}\) contains a cycle from \(S^{\prime}\). In this paper, we study minimal unavoidable sets of cycles in plane graphs of minimum degree at least 3 and present several graph constructions which forbid many cycle sets to be unavoidable. We also show the minimality of several small sets consisting of short cycles.

In this paper, we give two upper bounds for the extended energy of a graph one in terms of ordinary energy, maximum degree and minimum degree of a graph, and another bound in terms of forgotten index, inverse degree sum, order of a graph and minimum degree of a graph which improves an upper bound of Das et al. from [On spectral radius and energy of extended adjacency matrix of graphs, Appl. Math. Comput. 296 (2017), 116-123]. We present a pair of extended equienergetic graphs on \(n\) vertices for \(n\equiv 0(\text{mod } 8)\) starting with a pair of extended equienergetic non regular graphs on \(8\) vertices and also we construct a pair of extended equienergetic graphs on \(n\) vertices for all \(n\geq 9\) starting with a pair of equienergetic regular graphs on \(9\) vertices.

The paper continues the interesting study of the domination subdivision number and the domination multisubdivision number. On the basis of the constructive characterization of the trees with the domination subdivision number equal to 3 given in [H. Aram, S.M. Sheikholeslami, O. Favaron, Domination subdivision number of trees, Discrete Math. 309 (2009), 622-628], we constructively characterize all connected unicyclic graphs with the domination multisubdivision number equal to 3. We end with further questions and open problems.

Let \(G=(V,E)\) be a connected graph (or hypergraph) and let \(d(x,y)\) denote the distance between vertices \(x,y\in V(G)\). A subset \(W\subseteq V(G)\) is called a resolving set for \(G\) if for every pair of distinct vertices \(x,y\in V(G)\), there is \(w\in W\) such that \(d(x,w)\neq d(y,w)\). The minimum cardinality of a resolving set for \(G\) is called the metric dimension of \(G\), denoted by \(\beta(G)\). The circulant graph \(C_n(1,2,\ldots,t)\) has vertex set \(\{v_0,v_1,\ldots,v_{n-1}\}\) and edges \(v_iv_{i+j}\) where \(0\leq i\leq n-1\) and \(1\leq j\leq t\) and the indices are taken modulo \(n\) (\(2\leq t\leq\left\lfloor\frac{n}{2}\right\rfloor\)). In this paper we determine the exact metric dimension of the circulant graphs \(C_n(1,2,\ldots,t)\), extending previous results due to Borchert and Gosselin (2013), Grigorious et al. (2014), and Vetrik (2016). In particular, we show that \(\beta(C_n(1,2,\ldots,t))=\beta(C_{n+2t}(1,2,\ldots,t))\) for large enough \(n\), which implies that the metric dimension of these circulants is completely determined by the congruence class of \(n\) modulo \(2t\). We determine the exact value of \(\beta(C_n(1,2,\ldots,t))\) for \(n\equiv 2\bmod 2t\) and \(n\equiv (t+1)\bmod 2t\) and we give better bounds on the metric dimension of these circulants for \(n\equiv 0\bmod 2t\) and \(n\equiv 1 \bmod 2t\). In addition, we bound the metric dimension of Cartesian products of circulant graphs.

A \(k\)-total edge product cordial labeling is a variant of the well-known cordial labeling. In this paper we characterize connected graphs of order at least 15 admitting a 3-total edge product cordial labeling.

A simple graph \(G\) is called a compact graph if \(G\) contains no isolated vertices and for each pair \(x\), \(y\) of non-adjacent vertices of \(G\), there is a vertex \(z\) with \(N(x)\cup N(y)\subseteq N(z)\), where \(N(v)\) is the neighborhood of \(v\), for every vertex \(v\) of \(G\). In this paper, compact graphs with sufficient number of edges are studied. Also, it is proved that every regular compact graph is strongly regular. Some results about cycles in compact graphs are proved, too. Among other results, it is proved that if the ascending chain condition holds for the set of neighbors of a compact graph \(G\), then the descending chain condition holds for the set of neighbors of \(G\).

Given a graph \(G=(V,E)\), the subdivision of an edge \(e=uv\in E(G)\) means the substitution of the edge \(e\) by a vertex \(x\) and the new edges \(ux\) and \(xv\). The domination subdivision number of a graph \(G\) is the minimum number of edges of \(G\) which must be subdivided (where each edge can be subdivided at most once) in order to increase the domination number. Also, the domination multisubdivision number of \(G\) is the minimum number of subdivisions which must be done in one edge such that the domination number increases. Moreover, the concepts of paired domination and independent domination subdivision (respectively multisubdivision) numbers are defined similarly. In this paper we study the domination, paired domination and independent domination (subdivision and multisubdivision) numbers of the generalized corona graphs.

A \(2\)-rainbow dominating function of a graph \(G\left(V(G),E(G)\right)\) is a function \(f\) that assigns to each vertex a set of colors chosen from the set \(\{1,2\}\) so that for each vertex with \(f(v)=\emptyset\) we have \({\textstyle\bigcup_{u\in N(v)}} f(u)=\{1,2\}\). The weight of a \(2\)RDF \(f\) is defined as \(w\left( f\right)={\textstyle\sum\nolimits_{v\in V(G)}} |f(v)|\). The minimum weight of a \(2\)RDF is called the \(2\)-rainbow domination number of \(G\), denoted by \(\gamma_{2r}(G)\). The vertex criticality index of a \(2\)-rainbow domination of a graph \(G\) is defined as \(ci_{2r}^{v}(G)=(\sum\nolimits_{v\in V(G)}(\gamma_{2r}\left(G\right) -\gamma_{2r}\left( G-v\right)))/\left\vert V(G)\right\vert\), the edge removal criticality index of a \(2\)-rainbow domination of a graph \(G\) is defined as \(ci_{2r}^{-e}(G)=(\sum\nolimits_{e\in E(G)}(\gamma_{2r}\left(G\right)-\gamma_{2r}\left( G-e\right)))/\left\vert E(G)\right\vert\) and the edge addition of a \(2\)-rainbow domination criticality index of \(G\) is defined as \(ci_{2r}^{+e}(G)=(\sum\nolimits_{e\in E(\overline{G})}(\gamma_{2r}\left(G\right)-\gamma_{2r}\left( G+e\right)))/\left\vert E(\overline{G})\right\vert\), where \(\overline{G}\) is the complement graph of \(G\). In this paper, we determine the criticality indices of paths and cycles.